\documentclass[a4paper]{article}
\begin{document}
\title{Homework 2}
\author{5120719013 Feng Shi}
\date{2013.06.05}
\maketitle

\section{Ex 2.11}
Solution:
	
	The volume of a d-dimensional sphere with a constant radius r independent of d is
	$r^{d}V(d)$, where $V(d)$ stands for the volume of a unit-radius d-dimensional sphere.
	
	Assume $d$ is even and use Stirling's approximation for $\Gamma(\frac{d}{2})$ :
	\begin{displaymath}
		\Gamma(1+\frac{d}{2}) = (\frac{d}{2})! \sim (\frac{d}{2e})^{\frac{d}{2}} \sqrt{\pi d}
	\end{displaymath}
	So the volume is
	$$V_{d}(r) = \frac{\pi^{\frac{d}{2}} r^{d}}{(\frac{d}{2e})^\frac{d}{2} \sqrt{\pi d}}$$
	as assumed is a constant. Setting $V_{d}(r)=c$, yields
	$$r=\sqrt[d]{\frac{cd\Gamma(\frac{d}{2})}{2}}$$
	$$r=\sqrt[d]{c(\frac{d}{2\pi e})^{\frac{d}{2}}\sqrt{d\pi}}$$
	When $r=\sqrt{\frac{d}{2\pi e}}$, $V_d(r)$ is constant.

\section{Ex 2.15}
Solution:
	
	Fix $x_{1}=1$ on $x_{1}-axis$ the north pole, the volume of the cylinder entirely fixed 
	in the upper hemisphere has height $x_{1}$, surface area of the volume of a $(d-1)$-dimensional
	sphere with radius $\sqrt{1-x_{1}^{2}}$, that yields the volume of the cylinder :
	\begin{displaymath}
		V_{d}(x_{1}) = V(d-1) (1-x_{1}^{2})^{\frac{d-1}{2}} x_{1}
	\end{displaymath}
	where V(d-1) is a constant. Let $f(x) = (1-x^{2})^\frac{d-1}{2} x$.

	$V_{d}(x_{1})$ has maximium value when $x_{1} = \sqrt{\frac{1}{d}}$, this comes from setting
	the derivative of $f(x)$ equal to 0 :
	\begin{displaymath}
		f'(x) = (1-x^{2})^\frac{d-1}{2} - (d-1) x^{2} (1-x^{2})^\frac{d-3}{2} = 0
	\end{displaymath}
	which implies $x_{1} = \sqrt{\frac{1}{d}}$ .
\end{document}
